What is the advantage of using Entity bean for database operations, over directly using JDBC API to do database operations? When would I use one over the other?

Entity Beans actually represents the data in a database. It is not that Entity Beans replaces JDBC API. There are two types of Entity Beans Container Managed and Bean Mananged.

In Container Managed Entity Bean - Whenever the instance of the bean is created the container automatically retrieves the data from the DB/Persistance storage and assigns to the object variables in bean for user to manipulate or use them. For this the developer needs to map the fields in the database to the variables in deployment descriptor files (which varies for each vendor).

In the Bean Managed Entity Bean - The developer has to specifically make connection, retrive values, assign them to the objects in the ejbLoad() which will be called by the container when it instatiates a bean object. Similarly in the ejbStore() the container saves the object values back the the persistance storage. ejbLoad and ejbStore are callback methods and can be only invoked by the container.

Apart from this, when you use Entity beans you dont need to worry about database transaction handling, database connection pooling etc. which are taken care by the ejb container. But in case of JDBC you have to explicitly do the above features.